LMFDB - Embedded newform 3380.2.a.r.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3380,2,Mod(1,3380)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3380, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3380.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3380 = 2^{2} \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3380.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$26.9894358832$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 19x^{7} + 16x^{6} + 106x^{5} - 87x^{4} - 153x^{3} + 149x^{2} - 26x + 1 $ x^9 - x^8 - 19x^7 + 16x^6 + 106x^5 - 87x^4 - 153x^3 + 149x^2 - 26*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.40883$ of defining polynomial
Character	$\chi$	$=$	3380.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.40883 q^{3} -1.00000 q^{5} +2.17051 q^{7} -1.01520 q^{9} +O(q^{10})$	$q-1.40883 q^{3} -1.00000 q^{5} +2.17051 q^{7} -1.01520 q^{9} +5.73367 q^{11} +1.40883 q^{15} -5.24468 q^{17} +7.00804 q^{19} -3.05787 q^{21} -8.19819 q^{23} +1.00000 q^{25} +5.65673 q^{27} +9.54822 q^{29} +2.41917 q^{31} -8.07777 q^{33} -2.17051 q^{35} -7.16094 q^{37} -7.16158 q^{41} +7.76780 q^{43} +1.01520 q^{45} +5.41944 q^{47} -2.28890 q^{49} +7.38886 q^{51} +3.94223 q^{53} -5.73367 q^{55} -9.87313 q^{57} +0.993988 q^{59} -4.28702 q^{61} -2.20350 q^{63} -11.6322 q^{67} +11.5499 q^{69} -5.19073 q^{71} +3.06083 q^{73} -1.40883 q^{75} +12.4450 q^{77} +7.33131 q^{79} -4.92377 q^{81} -2.86146 q^{83} +5.24468 q^{85} -13.4518 q^{87} +12.2808 q^{89} -3.40820 q^{93} -7.00804 q^{95} +7.35295 q^{97} -5.82083 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - q^{3} - 9 q^{5} - q^{7} + 12 q^{9}+O(q^{10}) $ 9 * q - q^3 - 9 * q^5 - q^7 + 12 * q^9	$ 9 q - q^{3} - 9 q^{5} - q^{7} + 12 q^{9} + 7 q^{11} + q^{15} + 13 q^{17} + 4 q^{19} + 3 q^{21} + 12 q^{23} + 9 q^{25} - 4 q^{27} + 16 q^{29} - 13 q^{31} - 34 q^{33} + q^{35} - q^{37} + 6 q^{41} + q^{43} - 12 q^{45} + 2 q^{47} + 20 q^{49} + 11 q^{51} + 30 q^{53} - 7 q^{55} - 38 q^{57} - 15 q^{59} + 21 q^{61} + 17 q^{63} + 7 q^{67} + 15 q^{69} + 7 q^{71} + 28 q^{73} - q^{75} + 46 q^{77} + 31 q^{79} + 41 q^{81} - 45 q^{83} - 13 q^{85} + 28 q^{87} + 41 q^{89} + 11 q^{93} - 4 q^{95} - 8 q^{97} + 81 q^{99}+O(q^{100}) $ 9 * q - q^3 - 9 * q^5 - q^7 + 12 * q^9 + 7 * q^11 + q^15 + 13 * q^17 + 4 * q^19 + 3 * q^21 + 12 * q^23 + 9 * q^25 - 4 * q^27 + 16 * q^29 - 13 * q^31 - 34 * q^33 + q^35 - q^37 + 6 * q^41 + q^43 - 12 * q^45 + 2 * q^47 + 20 * q^49 + 11 * q^51 + 30 * q^53 - 7 * q^55 - 38 * q^57 - 15 * q^59 + 21 * q^61 + 17 * q^63 + 7 * q^67 + 15 * q^69 + 7 * q^71 + 28 * q^73 - q^75 + 46 * q^77 + 31 * q^79 + 41 * q^81 - 45 * q^83 - 13 * q^85 + 28 * q^87 + 41 * q^89 + 11 * q^93 - 4 * q^95 - 8 * q^97 + 81 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.40883	−0.813388	−0.406694	−	0.913564i	$-0.633318\pi$
$3$	−1.40883	−0.813388	−0.406694	+	0.913564i	$0.633318\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	2.17051	0.820374	0.410187	−	0.912001i	$-0.365463\pi$
$7$	2.17051	0.820374	0.410187	+	0.912001i	$0.365463\pi$
$8$	0	0
$9$	−1.01520	−0.338400
$10$	0	0
$11$	5.73367	1.72877	0.864384	−	0.502833i	$-0.167709\pi$
$11$	5.73367	1.72877	0.864384	+	0.502833i	$0.167709\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	1.40883	0.363758
$16$	0	0
$17$	−5.24468	−1.27202	−0.636011	−	0.771680i	$-0.719417\pi$
$17$	−5.24468	−1.27202	−0.636011	+	0.771680i	$0.719417\pi$
$18$	0	0
$19$	7.00804	1.60776	0.803878	−	0.594795i	$-0.202767\pi$
$19$	7.00804	1.60776	0.803878	+	0.594795i	$0.202767\pi$
$20$	0	0
$21$	−3.05787	−0.667283
$22$	0	0
$23$	−8.19819	−1.70944	−0.854721	−	0.519088i	$-0.826272\pi$
$23$	−8.19819	−1.70944	−0.854721	+	0.519088i	$0.826272\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.65673	1.08864
$28$	0	0
$29$	9.54822	1.77306	0.886530	−	0.462672i	$-0.153109\pi$
$29$	9.54822	1.77306	0.886530	+	0.462672i	$0.153109\pi$
$30$	0	0
$31$	2.41917	0.434496	0.217248	−	0.976116i	$-0.430292\pi$
$31$	2.41917	0.434496	0.217248	+	0.976116i	$0.430292\pi$
$32$	0	0
$33$	−8.07777	−1.40616
$34$	0	0
$35$	−2.17051	−0.366883
$36$	0	0
$37$	−7.16094	−1.17725	−0.588626	−	0.808406i	$-0.700331\pi$
$37$	−7.16094	−1.17725	−0.588626	+	0.808406i	$0.700331\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−7.16158	−1.11845	−0.559225	−	0.829016i	$-0.688901\pi$
$41$	−7.16158	−1.11845	−0.559225	+	0.829016i	$0.688901\pi$
$42$	0	0
$43$	7.76780	1.18458	0.592290	−	0.805725i	$-0.298224\pi$
$43$	7.76780	1.18458	0.592290	+	0.805725i	$0.298224\pi$
$44$	0	0
$45$	1.01520	0.151337
$46$	0	0
$47$	5.41944	0.790506	0.395253	−	0.918572i	$-0.370657\pi$
$47$	5.41944	0.790506	0.395253	+	0.918572i	$0.370657\pi$
$48$	0	0
$49$	−2.28890	−0.326986
$50$	0	0
$51$	7.38886	1.03465
$52$	0	0
$53$	3.94223	0.541508	0.270754	−	0.962649i	$-0.412727\pi$
$53$	3.94223	0.541508	0.270754	+	0.962649i	$0.412727\pi$
$54$	0	0
$55$	−5.73367	−0.773128
$56$	0	0
$57$	−9.87313	−1.30773
$58$	0	0
$59$	0.993988	0.129406	0.0647031	−	0.997905i	$-0.479390\pi$
$59$	0.993988	0.129406	0.0647031	+	0.997905i	$0.479390\pi$
$60$	0	0
$61$	−4.28702	−0.548897	−0.274448	−	0.961602i	$-0.588495\pi$
$61$	−4.28702	−0.548897	−0.274448	+	0.961602i	$0.588495\pi$
$62$	0	0
$63$	−2.20350	−0.277615
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−11.6322	−1.42110	−0.710548	−	0.703648i	$-0.751553\pi$
$67$	−11.6322	−1.42110	−0.710548	+	0.703648i	$0.751553\pi$
$68$	0	0
$69$	11.5499	1.39044
$70$	0	0
$71$	−5.19073	−0.616026	−0.308013	−	0.951382i	$-0.599664\pi$
$71$	−5.19073	−0.616026	−0.308013	+	0.951382i	$0.599664\pi$
$72$	0	0
$73$	3.06083	0.358243	0.179121	−	0.983827i	$-0.442674\pi$
$73$	3.06083	0.358243	0.179121	+	0.983827i	$0.442674\pi$
$74$	0	0
$75$	−1.40883	−0.162678
$76$	0	0
$77$	12.4450	1.41824
$78$	0	0
$79$	7.33131	0.824837	0.412419	−	0.910994i	$-0.364684\pi$
$79$	7.33131	0.824837	0.412419	+	0.910994i	$0.364684\pi$
$80$	0	0
$81$	−4.92377	−0.547085
$82$	0	0
$83$	−2.86146	−0.314086	−0.157043	−	0.987592i	$-0.550196\pi$
$83$	−2.86146	−0.314086	−0.157043	+	0.987592i	$0.550196\pi$
$84$	0	0
$85$	5.24468	0.568866
$86$	0	0
$87$	−13.4518	−1.44219
$88$	0	0
$89$	12.2808	1.30176	0.650881	−	0.759180i	$-0.274400\pi$
$89$	12.2808	1.30176	0.650881	+	0.759180i	$0.274400\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−3.40820	−0.353414
$94$	0	0
$95$	−7.00804	−0.719010
$96$	0	0
$97$	7.35295	0.746579	0.373289	−	0.927715i	$-0.378230\pi$
$97$	7.35295	0.746579	0.373289	+	0.927715i	$0.378230\pi$
$98$	0	0
$99$	−5.82083	−0.585015
$100$	0	0
$101$	−13.4407	−1.33740	−0.668699	−	0.743533i	$-0.733148\pi$
$101$	−13.4407	−1.33740	−0.668699	+	0.743533i	$0.733148\pi$
$102$	0	0
$103$	7.68074	0.756806	0.378403	−	0.925641i	$-0.376473\pi$
$103$	7.68074	0.756806	0.378403	+	0.925641i	$0.376473\pi$
$104$	0	0
$105$	3.05787	0.298418
$106$	0	0
$107$	−8.06583	−0.779754	−0.389877	−	0.920867i	$-0.627482\pi$
$107$	−8.06583	−0.779754	−0.389877	+	0.920867i	$0.627482\pi$
$108$	0	0
$109$	7.34155	0.703193	0.351597	−	0.936152i	$-0.385639\pi$
$109$	7.34155	0.703193	0.351597	+	0.936152i	$0.385639\pi$
$110$	0	0
$111$	10.0885	0.957562
$112$	0	0
$113$	−0.260722	−0.0245267	−0.0122633	−	0.999925i	$-0.503904\pi$
$113$	−0.260722	−0.0245267	−0.0122633	+	0.999925i	$0.503904\pi$
$114$	0	0
$115$	8.19819	0.764485
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−11.3836	−1.04353
$120$	0	0
$121$	21.8750	1.98864
$122$	0	0
$123$	10.0894	0.909734
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	12.6915	1.12619	0.563096	−	0.826392i	$-0.309610\pi$
$127$	12.6915	1.12619	0.563096	+	0.826392i	$0.309610\pi$
$128$	0	0
$129$	−10.9435	−0.963522
$130$	0	0
$131$	0.587367	0.0513185	0.0256592	−	0.999671i	$-0.491832\pi$
$131$	0.587367	0.0513185	0.0256592	+	0.999671i	$0.491832\pi$
$132$	0	0
$133$	15.2110	1.31896
$134$	0	0
$135$	−5.65673	−0.486854
$136$	0	0
$137$	−15.5251	−1.32640	−0.663200	−	0.748443i	$-0.730802\pi$
$137$	−15.5251	−1.32640	−0.663200	+	0.748443i	$0.730802\pi$
$138$	0	0
$139$	18.8204	1.59632	0.798161	−	0.602444i	$-0.205806\pi$
$139$	18.8204	1.59632	0.798161	+	0.602444i	$0.205806\pi$
$140$	0	0
$141$	−7.63506	−0.642988
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−9.54822	−0.792936
$146$	0	0
$147$	3.22467	0.265966
$148$	0	0
$149$	16.9576	1.38922	0.694609	−	0.719387i	$-0.255577\pi$
$149$	16.9576	1.38922	0.694609	+	0.719387i	$0.255577\pi$
$150$	0	0
$151$	10.9444	0.890640	0.445320	−	0.895371i	$-0.353090\pi$
$151$	10.9444	0.890640	0.445320	+	0.895371i	$0.353090\pi$
$152$	0	0
$153$	5.32440	0.430453
$154$	0	0
$155$	−2.41917	−0.194313
$156$	0	0
$157$	10.0706	0.803718	0.401859	−	0.915702i	$-0.368364\pi$
$157$	10.0706	0.803718	0.401859	+	0.915702i	$0.368364\pi$
$158$	0	0
$159$	−5.55394	−0.440456
$160$	0	0
$161$	−17.7942	−1.40238
$162$	0	0
$163$	19.5144	1.52849	0.764243	−	0.644928i	$-0.223113\pi$
$163$	19.5144	1.52849	0.764243	+	0.644928i	$0.223113\pi$
$164$	0	0
$165$	8.07777	0.628853
$166$	0	0
$167$	13.4061	1.03740	0.518698	−	0.854958i	$-0.326417\pi$
$167$	13.4061	1.03740	0.518698	+	0.854958i	$0.326417\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	−7.11457	−0.544064
$172$	0	0
$173$	15.1841	1.15443	0.577214	−	0.816593i	$-0.304140\pi$
$173$	15.1841	1.15443	0.577214	+	0.816593i	$0.304140\pi$
$174$	0	0
$175$	2.17051	0.164075
$176$	0	0
$177$	−1.40036	−0.105257
$178$	0	0
$179$	1.24513	0.0930655	0.0465327	−	0.998917i	$-0.485183\pi$
$179$	1.24513	0.0930655	0.0465327	+	0.998917i	$0.485183\pi$
$180$	0	0
$181$	−4.69219	−0.348768	−0.174384	−	0.984678i	$-0.555793\pi$
$181$	−4.69219	−0.348768	−0.174384	+	0.984678i	$0.555793\pi$
$182$	0	0
$183$	6.03968	0.446466
$184$	0	0
$185$	7.16094	0.526483
$186$	0	0
$187$	−30.0713	−2.19903
$188$	0	0
$189$	12.2780	0.893091
$190$	0	0
$191$	11.1007	0.803218	0.401609	−	0.915811i	$-0.368451\pi$
$191$	11.1007	0.803218	0.401609	+	0.915811i	$0.368451\pi$
$192$	0	0
$193$	6.30549	0.453879	0.226940	−	0.973909i	$-0.427128\pi$
$193$	6.30549	0.453879	0.226940	+	0.973909i	$0.427128\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	15.9703	1.13783	0.568917	−	0.822395i	$-0.307363\pi$
$197$	15.9703	1.13783	0.568917	+	0.822395i	$0.307363\pi$
$198$	0	0
$199$	−12.0075	−0.851191	−0.425596	−	0.904914i	$-0.639935\pi$
$199$	−12.0075	−0.851191	−0.425596	+	0.904914i	$0.639935\pi$
$200$	0	0
$201$	16.3878	1.15590
$202$	0	0
$203$	20.7245	1.45457
$204$	0	0
$205$	7.16158	0.500186
$206$	0	0
$207$	8.32281	0.578475
$208$	0	0
$209$	40.1818	2.77943
$210$	0	0
$211$	−1.02582	−0.0706203	−0.0353102	−	0.999376i	$-0.511242\pi$
$211$	−1.02582	−0.0706203	−0.0353102	+	0.999376i	$0.511242\pi$
$212$	0	0
$213$	7.31285	0.501068
$214$	0	0
$215$	−7.76780	−0.529760
$216$	0	0
$217$	5.25083	0.356450
$218$	0	0
$219$	−4.31219	−0.291391
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−2.17108	−0.145387	−0.0726933	−	0.997354i	$-0.523159\pi$
$223$	−2.17108	−0.145387	−0.0726933	+	0.997354i	$0.523159\pi$
$224$	0	0
$225$	−1.01520	−0.0676800
$226$	0	0
$227$	−6.48039	−0.430119	−0.215059	−	0.976601i	$-0.568994\pi$
$227$	−6.48039	−0.430119	−0.215059	+	0.976601i	$0.568994\pi$
$228$	0	0
$229$	25.8710	1.70960	0.854801	−	0.518955i	$-0.173679\pi$
$229$	25.8710	1.70960	0.854801	+	0.518955i	$0.173679\pi$
$230$	0	0
$231$	−17.5328	−1.15358
$232$	0	0
$233$	26.4139	1.73043	0.865216	−	0.501400i	$-0.167181\pi$
$233$	26.4139	1.73043	0.865216	+	0.501400i	$0.167181\pi$
$234$	0	0
$235$	−5.41944	−0.353525
$236$	0	0
$237$	−10.3286	−0.670913
$238$	0	0
$239$	0.157397	0.0101811	0.00509057	−	0.999987i	$-0.498380\pi$
$239$	0.157397	0.0101811	0.00509057	+	0.999987i	$0.498380\pi$
$240$	0	0
$241$	−11.5671	−0.745103	−0.372551	−	0.928012i	$-0.621517\pi$
$241$	−11.5671	−0.745103	−0.372551	+	0.928012i	$0.621517\pi$
$242$	0	0
$243$	−10.0334	−0.643646
$244$	0	0
$245$	2.28890	0.146232
$246$	0	0
$247$	0	0
$248$	0	0
$249$	4.03131	0.255474
$250$	0	0
$251$	−1.54408	−0.0974615	−0.0487307	−	0.998812i	$-0.515518\pi$
$251$	−1.54408	−0.0974615	−0.0487307	+	0.998812i	$0.515518\pi$
$252$	0	0
$253$	−47.0057	−2.95523
$254$	0	0
$255$	−7.38886	−0.462709
$256$	0	0
$257$	14.9633	0.933388	0.466694	−	0.884419i	$-0.345445\pi$
$257$	14.9633	0.933388	0.466694	+	0.884419i	$0.345445\pi$
$258$	0	0
$259$	−15.5429	−0.965787
$260$	0	0
$261$	−9.69335	−0.600003
$262$	0	0
$263$	−15.1383	−0.933470	−0.466735	−	0.884397i	$-0.654570\pi$
$263$	−15.1383	−0.933470	−0.466735	+	0.884397i	$0.654570\pi$
$264$	0	0
$265$	−3.94223	−0.242170
$266$	0	0
$267$	−17.3015	−1.05884
$268$	0	0
$269$	−9.14065	−0.557315	−0.278658	−	0.960390i	$-0.589889\pi$
$269$	−9.14065	−0.557315	−0.278658	+	0.960390i	$0.589889\pi$
$270$	0	0
$271$	4.39788	0.267152	0.133576	−	0.991039i	$-0.457354\pi$
$271$	4.39788	0.267152	0.133576	+	0.991039i	$0.457354\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	5.73367	0.345753
$276$	0	0
$277$	26.9216	1.61756	0.808780	−	0.588112i	$-0.200128\pi$
$277$	26.9216	1.61756	0.808780	+	0.588112i	$0.200128\pi$
$278$	0	0
$279$	−2.45594	−0.147034
$280$	0	0
$281$	−7.40886	−0.441975	−0.220988	−	0.975277i	$-0.570928\pi$
$281$	−7.40886	−0.441975	−0.220988	+	0.975277i	$0.570928\pi$
$282$	0	0
$283$	−31.5220	−1.87379	−0.936894	−	0.349613i	$-0.886313\pi$
$283$	−31.5220	−1.87379	−0.936894	+	0.349613i	$0.886313\pi$
$284$	0	0
$285$	9.87313	0.584834
$286$	0	0
$287$	−15.5443	−0.917549
$288$	0	0
$289$	10.5067	0.618042
$290$	0	0
$291$	−10.3590	−0.607258
$292$	0	0
$293$	−9.19191	−0.536997	−0.268499	−	0.963280i	$-0.586527\pi$
$293$	−9.19191	−0.536997	−0.268499	+	0.963280i	$0.586527\pi$
$294$	0	0
$295$	−0.993988	−0.0578722
$296$	0	0
$297$	32.4338	1.88200
$298$	0	0
$299$	0	0
$300$	0	0
$301$	16.8601	0.971798
$302$	0	0
$303$	18.9356	1.08782
$304$	0	0
$305$	4.28702	0.245474
$306$	0	0
$307$	−20.6454	−1.17830	−0.589148	−	0.808025i	$-0.700537\pi$
$307$	−20.6454	−1.17830	−0.589148	+	0.808025i	$0.700537\pi$
$308$	0	0
$309$	−10.8209	−0.615577
$310$	0	0
$311$	8.12701	0.460841	0.230420	−	0.973091i	$-0.425990\pi$
$311$	8.12701	0.460841	0.230420	+	0.973091i	$0.425990\pi$
$312$	0	0
$313$	−5.45768	−0.308486	−0.154243	−	0.988033i	$-0.549294\pi$
$313$	−5.45768	−0.308486	−0.154243	+	0.988033i	$0.549294\pi$
$314$	0	0
$315$	2.20350	0.124153
$316$	0	0
$317$	10.7778	0.605342	0.302671	−	0.953095i	$-0.402122\pi$
$317$	10.7778	0.605342	0.302671	+	0.953095i	$0.402122\pi$
$318$	0	0
$319$	54.7464	3.06521
$320$	0	0
$321$	11.3634	0.634242
$322$	0	0
$323$	−36.7550	−2.04510
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−10.3430	−0.571969
$328$	0	0
$329$	11.7629	0.648511
$330$	0	0
$331$	−25.9764	−1.42779	−0.713896	−	0.700252i	$-0.753071\pi$
$331$	−25.9764	−1.42779	−0.713896	+	0.700252i	$0.753071\pi$
$332$	0	0
$333$	7.26979	0.398382
$334$	0	0
$335$	11.6322	0.635534
$336$	0	0
$337$	20.6275	1.12365	0.561824	−	0.827256i	$-0.310100\pi$
$337$	20.6275	1.12365	0.561824	+	0.827256i	$0.310100\pi$
$338$	0	0
$339$	0.367313	0.0199497
$340$	0	0
$341$	13.8707	0.751143
$342$	0	0
$343$	−20.1616	−1.08863
$344$	0	0
$345$	−11.5499	−0.621823
$346$	0	0
$347$	18.9341	1.01643	0.508217	−	0.861229i	$-0.330305\pi$
$347$	18.9341	1.01643	0.508217	+	0.861229i	$0.330305\pi$
$348$	0	0
$349$	4.53125	0.242552	0.121276	−	0.992619i	$-0.461301\pi$
$349$	4.53125	0.242552	0.121276	+	0.992619i	$0.461301\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−18.8925	−1.00554	−0.502772	−	0.864419i	$-0.667686\pi$
$353$	−18.8925	−1.00554	−0.502772	+	0.864419i	$0.667686\pi$
$354$	0	0
$355$	5.19073	0.275495
$356$	0	0
$357$	16.0376	0.848799
$358$	0	0
$359$	20.4677	1.08024	0.540122	−	0.841587i	$-0.318378\pi$
$359$	20.4677	1.08024	0.540122	+	0.841587i	$0.318378\pi$
$360$	0	0
$361$	30.1127	1.58488
$362$	0	0
$363$	−30.8181	−1.61753
$364$	0	0
$365$	−3.06083	−0.160211
$366$	0	0
$367$	22.6792	1.18384	0.591922	−	0.805995i	$-0.298369\pi$
$367$	22.6792	1.18384	0.591922	+	0.805995i	$0.298369\pi$
$368$	0	0
$369$	7.27044	0.378484
$370$	0	0
$371$	8.55665	0.444239
$372$	0	0
$373$	14.9049	0.771749	0.385874	−	0.922551i	$-0.373900\pi$
$373$	14.9049	0.771749	0.385874	+	0.922551i	$0.373900\pi$
$374$	0	0
$375$	1.40883	0.0727516
$376$	0	0
$377$	0	0
$378$	0	0
$379$	15.7604	0.809555	0.404778	−	0.914415i	$-0.367349\pi$
$379$	15.7604	0.809555	0.404778	+	0.914415i	$0.367349\pi$
$380$	0	0
$381$	−17.8802	−0.916030
$382$	0	0
$383$	1.21077	0.0618675	0.0309338	−	0.999521i	$-0.490152\pi$
$383$	1.21077	0.0618675	0.0309338	+	0.999521i	$0.490152\pi$
$384$	0	0
$385$	−12.4450	−0.634255
$386$	0	0
$387$	−7.88588	−0.400862
$388$	0	0
$389$	−30.4592	−1.54434	−0.772172	−	0.635414i	$-0.780829\pi$
$389$	−30.4592	−1.54434	−0.772172	+	0.635414i	$0.780829\pi$
$390$	0	0
$391$	42.9969	2.17445
$392$	0	0
$393$	−0.827500	−0.0417418
$394$	0	0
$395$	−7.33131	−0.368878
$396$	0	0
$397$	−30.6666	−1.53911	−0.769557	−	0.638578i	$-0.779523\pi$
$397$	−30.6666	−1.53911	−0.769557	+	0.638578i	$0.779523\pi$
$398$	0	0
$399$	−21.4297	−1.07283
$400$	0	0
$401$	−12.3885	−0.618652	−0.309326	−	0.950956i	$-0.600104\pi$
$401$	−12.3885	−0.618652	−0.309326	+	0.950956i	$0.600104\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	4.92377	0.244664
$406$	0	0
$407$	−41.0585	−2.03519
$408$	0	0
$409$	9.76935	0.483063	0.241532	−	0.970393i	$-0.422350\pi$
$409$	9.76935	0.483063	0.241532	+	0.970393i	$0.422350\pi$
$410$	0	0
$411$	21.8722	1.07888
$412$	0	0
$413$	2.15746	0.106162
$414$	0	0
$415$	2.86146	0.140464
$416$	0	0
$417$	−26.5147	−1.29843
$418$	0	0
$419$	5.65383	0.276208	0.138104	−	0.990418i	$-0.455899\pi$
$419$	5.65383	0.276208	0.138104	+	0.990418i	$0.455899\pi$
$420$	0	0
$421$	17.8610	0.870494	0.435247	−	0.900311i	$-0.356661\pi$
$421$	17.8610	0.870494	0.435247	+	0.900311i	$0.356661\pi$
$422$	0	0
$423$	−5.50182	−0.267507
$424$	0	0
$425$	−5.24468	−0.254405
$426$	0	0
$427$	−9.30501	−0.450301
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−27.3930	−1.31947	−0.659737	−	0.751497i	$-0.729332\pi$
$431$	−27.3930	−1.31947	−0.659737	+	0.751497i	$0.729332\pi$
$432$	0	0
$433$	−10.0057	−0.480841	−0.240421	−	0.970669i	$-0.577285\pi$
$433$	−10.0057	−0.480841	−0.240421	+	0.970669i	$0.577285\pi$
$434$	0	0
$435$	13.4518	0.644965
$436$	0	0
$437$	−57.4533	−2.74836
$438$	0	0
$439$	−1.38320	−0.0660167	−0.0330084	−	0.999455i	$-0.510509\pi$
$439$	−1.38320	−0.0660167	−0.0330084	+	0.999455i	$0.510509\pi$
$440$	0	0
$441$	2.32369	0.110652
$442$	0	0
$443$	15.6990	0.745883	0.372941	−	0.927855i	$-0.378349\pi$
$443$	15.6990	0.745883	0.372941	+	0.927855i	$0.378349\pi$
$444$	0	0
$445$	−12.2808	−0.582166
$446$	0	0
$447$	−23.8903	−1.12997
$448$	0	0
$449$	29.7220	1.40267	0.701335	−	0.712832i	$-0.252588\pi$
$449$	29.7220	1.40267	0.701335	+	0.712832i	$0.252588\pi$
$450$	0	0
$451$	−41.0622	−1.93354
$452$	0	0
$453$	−15.4187	−0.724436
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−0.208226	−0.00974041	−0.00487021	−	0.999988i	$-0.501550\pi$
$457$	−0.208226	−0.00974041	−0.00487021	+	0.999988i	$0.501550\pi$
$458$	0	0
$459$	−29.6678	−1.38477
$460$	0	0
$461$	6.24475	0.290847	0.145424	−	0.989369i	$-0.453546\pi$
$461$	6.24475	0.290847	0.145424	+	0.989369i	$0.453546\pi$
$462$	0	0
$463$	−3.42664	−0.159249	−0.0796246	−	0.996825i	$-0.525372\pi$
$463$	−3.42664	−0.159249	−0.0796246	+	0.996825i	$0.525372\pi$
$464$	0	0
$465$	3.40820	0.158052
$466$	0	0
$467$	26.2665	1.21547	0.607733	−	0.794141i	$-0.292079\pi$
$467$	26.2665	1.21547	0.607733	+	0.794141i	$0.292079\pi$
$468$	0	0
$469$	−25.2477	−1.16583
$470$	0	0
$471$	−14.1877	−0.653735
$472$	0	0
$473$	44.5380	2.04786
$474$	0	0
$475$	7.00804	0.321551
$476$	0	0
$477$	−4.00216	−0.183246
$478$	0	0
$479$	−3.46962	−0.158531	−0.0792654	−	0.996854i	$-0.525257\pi$
$479$	−3.46962	−0.158531	−0.0792654	+	0.996854i	$0.525257\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	25.0690	1.14068
$484$	0	0
$485$	−7.35295	−0.333880
$486$	0	0
$487$	−16.6306	−0.753606	−0.376803	−	0.926293i	$-0.622977\pi$
$487$	−16.6306	−0.753606	−0.376803	+	0.926293i	$0.622977\pi$
$488$	0	0
$489$	−27.4925	−1.24325
$490$	0	0
$491$	19.0416	0.859334	0.429667	−	0.902987i	$-0.358631\pi$
$491$	19.0416	0.859334	0.429667	+	0.902987i	$0.358631\pi$
$492$	0	0
$493$	−50.0774	−2.25537
$494$	0	0
$495$	5.82083	0.261627
$496$	0	0
$497$	−11.2665	−0.505372
$498$	0	0
$499$	34.4954	1.54423	0.772114	−	0.635485i	$-0.219200\pi$
$499$	34.4954	1.54423	0.772114	+	0.635485i	$0.219200\pi$
$500$	0	0
$501$	−18.8869	−0.843805
$502$	0	0
$503$	−23.5640	−1.05067	−0.525333	−	0.850897i	$-0.676059\pi$
$503$	−23.5640	−1.05067	−0.525333	+	0.850897i	$0.676059\pi$
$504$	0	0
$505$	13.4407	0.598102
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−25.8268	−1.14475	−0.572377	−	0.819991i	$-0.693978\pi$
$509$	−25.8268	−1.14475	−0.572377	+	0.819991i	$0.693978\pi$
$510$	0	0
$511$	6.64355	0.293893
$512$	0	0
$513$	39.6426	1.75026
$514$	0	0
$515$	−7.68074	−0.338454
$516$	0	0
$517$	31.0733	1.36660
$518$	0	0
$519$	−21.3918	−0.938998
$520$	0	0
$521$	−24.7194	−1.08298	−0.541489	−	0.840708i	$-0.682139\pi$
$521$	−24.7194	−1.08298	−0.541489	+	0.840708i	$0.682139\pi$
$522$	0	0
$523$	−22.6317	−0.989613	−0.494807	−	0.869003i	$-0.664761\pi$
$523$	−22.6317	−0.989613	−0.494807	+	0.869003i	$0.664761\pi$
$524$	0	0
$525$	−3.05787	−0.133457
$526$	0	0
$527$	−12.6878	−0.552689
$528$	0	0
$529$	44.2104	1.92219
$530$	0	0
$531$	−1.00910	−0.0437911
$532$	0	0
$533$	0	0
$534$	0	0
$535$	8.06583	0.348716
$536$	0	0
$537$	−1.75418	−0.0756983
$538$	0	0
$539$	−13.1238	−0.565282
$540$	0	0
$541$	16.2447	0.698413	0.349206	−	0.937046i	$-0.386451\pi$
$541$	16.2447	0.698413	0.349206	+	0.937046i	$0.386451\pi$
$542$	0	0
$543$	6.61049	0.283683
$544$	0	0
$545$	−7.34155	−0.314478
$546$	0	0
$547$	−24.2073	−1.03503	−0.517515	−	0.855674i	$-0.673143\pi$
$547$	−24.2073	−1.03503	−0.517515	+	0.855674i	$0.673143\pi$
$548$	0	0
$549$	4.35219	0.185747
$550$	0	0
$551$	66.9143	2.85065
$552$	0	0
$553$	15.9127	0.676675
$554$	0	0
$555$	−10.0885	−0.428235
$556$	0	0
$557$	16.6266	0.704492	0.352246	−	0.935907i	$-0.385418\pi$
$557$	16.6266	0.704492	0.352246	+	0.935907i	$0.385418\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	42.3653	1.78867
$562$	0	0
$563$	−2.94274	−0.124022	−0.0620109	−	0.998075i	$-0.519751\pi$
$563$	−2.94274	−0.124022	−0.0620109	+	0.998075i	$0.519751\pi$
$564$	0	0
$565$	0.260722	0.0109687
$566$	0	0
$567$	−10.6871	−0.448815
$568$	0	0
$569$	−45.8747	−1.92317	−0.961583	−	0.274514i	$-0.911483\pi$
$569$	−45.8747	−1.92317	−0.961583	+	0.274514i	$0.911483\pi$
$570$	0	0
$571$	−2.07821	−0.0869705	−0.0434853	−	0.999054i	$-0.513846\pi$
$571$	−2.07821	−0.0869705	−0.0434853	+	0.999054i	$0.513846\pi$
$572$	0	0
$573$	−15.6390	−0.653328
$574$	0	0
$575$	−8.19819	−0.341888
$576$	0	0
$577$	−42.5167	−1.76999	−0.884997	−	0.465597i	$-0.845839\pi$
$577$	−42.5167	−1.76999	−0.884997	+	0.465597i	$0.845839\pi$
$578$	0	0
$579$	−8.88336	−0.369180
$580$	0	0
$581$	−6.21082	−0.257668
$582$	0	0
$583$	22.6035	0.936141
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−8.29519	−0.342379	−0.171190	−	0.985238i	$-0.554761\pi$
$587$	−8.29519	−0.342379	−0.171190	+	0.985238i	$0.554761\pi$
$588$	0	0
$589$	16.9537	0.698563
$590$	0	0
$591$	−22.4994	−0.925500
$592$	0	0
$593$	−20.8549	−0.856407	−0.428203	−	0.903682i	$-0.640853\pi$
$593$	−20.8549	−0.856407	−0.428203	+	0.903682i	$0.640853\pi$
$594$	0	0
$595$	11.3836	0.466683
$596$	0	0
$597$	16.9166	0.692349
$598$	0	0
$599$	−19.1748	−0.783461	−0.391730	−	0.920080i	$-0.628123\pi$
$599$	−19.1748	−0.783461	−0.391730	+	0.920080i	$0.628123\pi$
$600$	0	0
$601$	46.9333	1.91445	0.957225	−	0.289345i	$-0.0934375\pi$
$601$	46.9333	1.91445	0.957225	+	0.289345i	$0.0934375\pi$
$602$	0	0
$603$	11.8090	0.480899
$604$	0	0
$605$	−21.8750	−0.889345
$606$	0	0
$607$	7.58350	0.307805	0.153902	−	0.988086i	$-0.450816\pi$
$607$	7.58350	0.307805	0.153902	+	0.988086i	$0.450816\pi$
$608$	0	0
$609$	−29.1972	−1.18313
$610$	0	0
$611$	0	0
$612$	0	0
$613$	14.1514	0.571569	0.285785	−	0.958294i	$-0.407746\pi$
$613$	14.1514	0.571569	0.285785	+	0.958294i	$0.407746\pi$
$614$	0	0
$615$	−10.0894	−0.406846
$616$	0	0
$617$	−13.2765	−0.534493	−0.267246	−	0.963628i	$-0.586114\pi$
$617$	−13.2765	−0.534493	−0.267246	+	0.963628i	$0.586114\pi$
$618$	0	0
$619$	28.0262	1.12647	0.563234	−	0.826298i	$-0.309557\pi$
$619$	28.0262	1.12647	0.563234	+	0.826298i	$0.309557\pi$
$620$	0	0
$621$	−46.3750	−1.86096
$622$	0	0
$623$	26.6556	1.06793
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−56.6093	−2.26076
$628$	0	0
$629$	37.5569	1.49749
$630$	0	0
$631$	−18.1132	−0.721074	−0.360537	−	0.932745i	$-0.617406\pi$
$631$	−18.1132	−0.721074	−0.360537	+	0.932745i	$0.617406\pi$
$632$	0	0
$633$	1.44520	0.0574417
$634$	0	0
$635$	−12.6915	−0.503648
$636$	0	0
$637$	0	0
$638$	0	0
$639$	5.26963	0.208463
$640$	0	0
$641$	11.1409	0.440038	0.220019	−	0.975496i	$-0.429388\pi$
$641$	11.1409	0.440038	0.220019	+	0.975496i	$0.429388\pi$
$642$	0	0
$643$	−4.70617	−0.185593	−0.0927965	−	0.995685i	$-0.529581\pi$
$643$	−4.70617	−0.185593	−0.0927965	+	0.995685i	$0.529581\pi$
$644$	0	0
$645$	10.9435	0.430900
$646$	0	0
$647$	6.21648	0.244395	0.122197	−	0.992506i	$-0.461006\pi$
$647$	6.21648	0.244395	0.122197	+	0.992506i	$0.461006\pi$
$648$	0	0
$649$	5.69920	0.223713
$650$	0	0
$651$	−7.39752	−0.289932
$652$	0	0
$653$	33.0832	1.29465	0.647323	−	0.762215i	$-0.275888\pi$
$653$	33.0832	1.29465	0.647323	+	0.762215i	$0.275888\pi$
$654$	0	0
$655$	−0.587367	−0.0229503
$656$	0	0
$657$	−3.10735	−0.121229
$658$	0	0
$659$	−10.1051	−0.393640	−0.196820	−	0.980440i	$-0.563061\pi$
$659$	−10.1051	−0.393640	−0.196820	+	0.980440i	$0.563061\pi$
$660$	0	0
$661$	−28.5808	−1.11166	−0.555832	−	0.831295i	$-0.687600\pi$
$661$	−28.5808	−1.11166	−0.555832	+	0.831295i	$0.687600\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−15.2110	−0.589857
$666$	0	0
$667$	−78.2781	−3.03094
$668$	0	0
$669$	3.05869	0.118256
$670$	0	0
$671$	−24.5804	−0.948915
$672$	0	0
$673$	−27.0625	−1.04318	−0.521590	−	0.853196i	$-0.674661\pi$
$673$	−27.0625	−1.04318	−0.521590	+	0.853196i	$0.674661\pi$
$674$	0	0
$675$	5.65673	0.217728
$676$	0	0
$677$	−48.4114	−1.86060	−0.930301	−	0.366798i	$-0.880454\pi$
$677$	−48.4114	−1.86060	−0.930301	+	0.366798i	$0.880454\pi$
$678$	0	0
$679$	15.9596	0.612474
$680$	0	0
$681$	9.12976	0.349853
$682$	0	0
$683$	20.6854	0.791507	0.395753	−	0.918357i	$-0.370484\pi$
$683$	20.6854	0.791507	0.395753	+	0.918357i	$0.370484\pi$
$684$	0	0
$685$	15.5251	0.593184
$686$	0	0
$687$	−36.4478	−1.39057
$688$	0	0
$689$	0	0
$690$	0	0
$691$	28.3098	1.07696	0.538478	−	0.842639i	$-0.318999\pi$
$691$	28.3098	1.07696	0.538478	+	0.842639i	$0.318999\pi$
$692$	0	0
$693$	−12.6341	−0.479931
$694$	0	0
$695$	−18.8204	−0.713897
$696$	0	0
$697$	37.5602	1.42269
$698$	0	0
$699$	−37.2127	−1.40751
$700$	0	0
$701$	−12.7481	−0.481490	−0.240745	−	0.970588i	$-0.577392\pi$
$701$	−12.7481	−0.481490	−0.240745	+	0.970588i	$0.577392\pi$
$702$	0	0
$703$	−50.1842	−1.89273
$704$	0	0
$705$	7.63506	0.287553
$706$	0	0
$707$	−29.1731	−1.09717
$708$	0	0
$709$	−44.3115	−1.66415	−0.832077	−	0.554660i	$-0.812848\pi$
$709$	−44.3115	−1.66415	−0.832077	+	0.554660i	$0.812848\pi$
$710$	0	0
$711$	−7.44275	−0.279125
$712$	0	0
$713$	−19.8328	−0.742746
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−0.221745	−0.00828122
$718$	0	0
$719$	7.49703	0.279592	0.139796	−	0.990180i	$-0.455355\pi$
$719$	7.49703	0.279592	0.139796	+	0.990180i	$0.455355\pi$
$720$	0	0
$721$	16.6711	0.620864
$722$	0	0
$723$	16.2961	0.606058
$724$	0	0
$725$	9.54822	0.354612
$726$	0	0
$727$	23.3642	0.866530	0.433265	−	0.901267i	$-0.357361\pi$
$727$	23.3642	0.866530	0.433265	+	0.901267i	$0.357361\pi$
$728$	0	0
$729$	28.9067	1.07062
$730$	0	0
$731$	−40.7397	−1.50681
$732$	0	0
$733$	27.2189	1.00535	0.502677	−	0.864474i	$-0.332348\pi$
$733$	27.2189	1.00535	0.502677	+	0.864474i	$0.332348\pi$
$734$	0	0
$735$	−3.22467	−0.118944
$736$	0	0
$737$	−66.6951	−2.45675
$738$	0	0
$739$	−35.2389	−1.29628	−0.648141	−	0.761520i	$-0.724453\pi$
$739$	−35.2389	−1.29628	−0.648141	+	0.761520i	$0.724453\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	9.76023	0.358068	0.179034	−	0.983843i	$-0.442703\pi$
$743$	9.76023	0.358068	0.179034	+	0.983843i	$0.442703\pi$
$744$	0	0
$745$	−16.9576	−0.621277
$746$	0	0
$747$	2.90496	0.106287
$748$	0	0
$749$	−17.5069	−0.639690
$750$	0	0
$751$	53.1333	1.93886	0.969431	−	0.245363i	$-0.0789073\pi$
$751$	53.1333	1.93886	0.969431	+	0.245363i	$0.0789073\pi$
$752$	0	0
$753$	2.17534	0.0792740
$754$	0	0
$755$	−10.9444	−0.398306
$756$	0	0
$757$	−33.6611	−1.22343	−0.611716	−	0.791077i	$-0.709520\pi$
$757$	−33.6611	−1.22343	−0.611716	+	0.791077i	$0.709520\pi$
$758$	0	0
$759$	66.2231	2.40375
$760$	0	0
$761$	38.2985	1.38832	0.694160	−	0.719821i	$-0.255776\pi$
$761$	38.2985	1.38832	0.694160	+	0.719821i	$0.255776\pi$
$762$	0	0
$763$	15.9349	0.576882
$764$	0	0
$765$	−5.32440	−0.192504
$766$	0	0
$767$	0	0
$768$	0	0
$769$	28.9380	1.04353	0.521765	−	0.853089i	$-0.325274\pi$
$769$	28.9380	1.04353	0.521765	+	0.853089i	$0.325274\pi$
$770$	0	0
$771$	−21.0808	−0.759206
$772$	0	0
$773$	−22.4859	−0.808762	−0.404381	−	0.914591i	$-0.632513\pi$
$773$	−22.4859	−0.808762	−0.404381	+	0.914591i	$0.632513\pi$
$774$	0	0
$775$	2.41917	0.0868992
$776$	0	0
$777$	21.8972	0.785559
$778$	0	0
$779$	−50.1887	−1.79820
$780$	0	0
$781$	−29.7619	−1.06497
$782$	0	0
$783$	54.0117	1.93022
$784$	0	0
$785$	−10.0706	−0.359434
$786$	0	0
$787$	−16.0779	−0.573117	−0.286558	−	0.958063i	$-0.592511\pi$
$787$	−16.0779	−0.573117	−0.286558	+	0.958063i	$0.592511\pi$
$788$	0	0
$789$	21.3273	0.759273
$790$	0	0
$791$	−0.565899	−0.0201211
$792$	0	0
$793$	0	0
$794$	0	0
$795$	5.55394	0.196978
$796$	0	0
$797$	−51.8604	−1.83699	−0.918494	−	0.395434i	$-0.870594\pi$
$797$	−51.8604	−1.83699	−0.918494	+	0.395434i	$0.870594\pi$
$798$	0	0
$799$	−28.4232	−1.00554
$800$	0	0
$801$	−12.4675	−0.440516
$802$	0	0
$803$	17.5498	0.619319
$804$	0	0
$805$	17.7942	0.627164
$806$	0	0
$807$	12.8776	0.453314
$808$	0	0
$809$	26.9372	0.947060	0.473530	−	0.880778i	$-0.342980\pi$
$809$	26.9372	0.947060	0.473530	+	0.880778i	$0.342980\pi$
$810$	0	0
$811$	−36.5575	−1.28371	−0.641854	−	0.766827i	$-0.721834\pi$
$811$	−36.5575	−1.28371	−0.641854	+	0.766827i	$0.721834\pi$
$812$	0	0
$813$	−6.19586	−0.217298
$814$	0	0
$815$	−19.5144	−0.683560
$816$	0	0
$817$	54.4371	1.90451
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−2.59771	−0.0906607	−0.0453304	−	0.998972i	$-0.514434\pi$
$821$	−2.59771	−0.0906607	−0.0453304	+	0.998972i	$0.514434\pi$
$822$	0	0
$823$	−7.85046	−0.273650	−0.136825	−	0.990595i	$-0.543690\pi$
$823$	−7.85046	−0.273650	−0.136825	+	0.990595i	$0.543690\pi$
$824$	0	0
$825$	−8.07777	−0.281232
$826$	0	0
$827$	−7.12878	−0.247892	−0.123946	−	0.992289i	$-0.539555\pi$
$827$	−7.12878	−0.247892	−0.123946	+	0.992289i	$0.539555\pi$
$828$	0	0
$829$	51.3687	1.78411	0.892054	−	0.451929i	$-0.149264\pi$
$829$	51.3687	1.78411	0.892054	+	0.451929i	$0.149264\pi$
$830$	0	0
$831$	−37.9279	−1.31570
$832$	0	0
$833$	12.0046	0.415933
$834$	0	0
$835$	−13.4061	−0.463938
$836$	0	0
$837$	13.6846	0.473009
$838$	0	0
$839$	−0.322289	−0.0111266	−0.00556332	−	0.999985i	$-0.501771\pi$
$839$	−0.322289	−0.0111266	−0.00556332	+	0.999985i	$0.501771\pi$
$840$	0	0
$841$	62.1685	2.14374
$842$	0	0
$843$	10.4378	0.359497
$844$	0	0
$845$	0	0
$846$	0	0
$847$	47.4798	1.63143
$848$	0	0
$849$	44.4091	1.52412
$850$	0	0
$851$	58.7068	2.01244
$852$	0	0
$853$	−18.1009	−0.619763	−0.309881	−	0.950775i	$-0.600289\pi$
$853$	−18.1009	−0.619763	−0.309881	+	0.950775i	$0.600289\pi$
$854$	0	0
$855$	7.11457	0.243313
$856$	0	0
$857$	−44.3079	−1.51353	−0.756764	−	0.653688i	$-0.773221\pi$
$857$	−44.3079	−1.51353	−0.756764	+	0.653688i	$0.773221\pi$
$858$	0	0
$859$	14.5314	0.495807	0.247903	−	0.968785i	$-0.420258\pi$
$859$	14.5314	0.495807	0.247903	+	0.968785i	$0.420258\pi$
$860$	0	0
$861$	21.8992	0.746323
$862$	0	0
$863$	23.5161	0.800496	0.400248	−	0.916407i	$-0.368924\pi$
$863$	23.5161	0.800496	0.400248	+	0.916407i	$0.368924\pi$
$864$	0	0
$865$	−15.1841	−0.516276
$866$	0	0
$867$	−14.8022	−0.502707
$868$	0	0
$869$	42.0354	1.42595
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−7.46471	−0.252642
$874$	0	0
$875$	−2.17051	−0.0733765
$876$	0	0
$877$	−27.6427	−0.933428	−0.466714	−	0.884408i	$-0.654562\pi$
$877$	−27.6427	−0.933428	−0.466714	+	0.884408i	$0.654562\pi$
$878$	0	0
$879$	12.9498	0.436787
$880$	0	0
$881$	−11.7134	−0.394635	−0.197318	−	0.980340i	$-0.563223\pi$
$881$	−11.7134	−0.394635	−0.197318	+	0.980340i	$0.563223\pi$
$882$	0	0
$883$	−5.53015	−0.186105	−0.0930523	−	0.995661i	$-0.529662\pi$
$883$	−5.53015	−0.186105	−0.0930523	+	0.995661i	$0.529662\pi$
$884$	0	0
$885$	1.40036	0.0470726
$886$	0	0
$887$	−15.0601	−0.505668	−0.252834	−	0.967510i	$-0.581363\pi$
$887$	−15.0601	−0.505668	−0.252834	+	0.967510i	$0.581363\pi$
$888$	0	0
$889$	27.5471	0.923899
$890$	0	0
$891$	−28.2313	−0.945783
$892$	0	0
$893$	37.9797	1.27094
$894$	0	0
$895$	−1.24513	−0.0416202
$896$	0	0
$897$	0	0
$898$	0	0
$899$	23.0988	0.770388
$900$	0	0
$901$	−20.6758	−0.688810
$902$	0	0
$903$	−23.7530	−0.790449
$904$	0	0
$905$	4.69219	0.155974
$906$	0	0
$907$	47.0337	1.56173	0.780865	−	0.624700i	$-0.214779\pi$
$907$	47.0337	1.56173	0.780865	+	0.624700i	$0.214779\pi$
$908$	0	0
$909$	13.6450	0.452575
$910$	0	0
$911$	18.2632	0.605086	0.302543	−	0.953136i	$-0.402164\pi$
$911$	18.2632	0.605086	0.302543	+	0.953136i	$0.402164\pi$
$912$	0	0
$913$	−16.4067	−0.542982
$914$	0	0
$915$	−6.03968	−0.199666
$916$	0	0
$917$	1.27488	0.0421004
$918$	0	0
$919$	23.9725	0.790780	0.395390	−	0.918513i	$-0.370610\pi$
$919$	23.9725	0.790780	0.395390	+	0.918513i	$0.370610\pi$
$920$	0	0
$921$	29.0859	0.958412
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−7.16094	−0.235450
$926$	0	0
$927$	−7.79749	−0.256103
$928$	0	0
$929$	−31.2931	−1.02669	−0.513347	−	0.858181i	$-0.671595\pi$
$929$	−31.2931	−1.02669	−0.513347	+	0.858181i	$0.671595\pi$
$930$	0	0
$931$	−16.0407	−0.525713
$932$	0	0
$933$	−11.4496	−0.374842
$934$	0	0
$935$	30.0713	0.983437
$936$	0	0
$937$	−0.100918	−0.00329684	−0.00164842	−	0.999999i	$-0.500525\pi$
$937$	−0.100918	−0.00329684	−0.00164842	+	0.999999i	$0.500525\pi$
$938$	0	0
$939$	7.68894	0.250919
$940$	0	0
$941$	−32.1209	−1.04711	−0.523555	−	0.851992i	$-0.675395\pi$
$941$	−32.1209	−1.04711	−0.523555	+	0.851992i	$0.675395\pi$
$942$	0	0
$943$	58.7120	1.91193
$944$	0	0
$945$	−12.2780	−0.399403
$946$	0	0
$947$	−11.3926	−0.370209	−0.185104	−	0.982719i	$-0.559262\pi$
$947$	−11.3926	−0.370209	−0.185104	+	0.982719i	$0.559262\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−15.1841	−0.492378
$952$	0	0
$953$	20.0389	0.649125	0.324562	−	0.945864i	$-0.394783\pi$
$953$	20.0389	0.649125	0.324562	+	0.945864i	$0.394783\pi$
$954$	0	0
$955$	−11.1007	−0.359210
$956$	0	0
$957$	−77.1283	−2.49320
$958$	0	0
$959$	−33.6973	−1.08814
$960$	0	0
$961$	−25.1476	−0.811213
$962$	0	0
$963$	8.18844	0.263869
$964$	0	0
$965$	−6.30549	−0.202981
$966$	0	0
$967$	−17.7835	−0.571878	−0.285939	−	0.958248i	$-0.592306\pi$
$967$	−17.7835	−0.571878	−0.285939	+	0.958248i	$0.592306\pi$
$968$	0	0
$969$	51.7815	1.66346
$970$	0	0
$971$	10.0669	0.323062	0.161531	−	0.986868i	$-0.448357\pi$
$971$	10.0669	0.323062	0.161531	+	0.986868i	$0.448357\pi$
$972$	0	0
$973$	40.8497	1.30958
$974$	0	0
$975$	0	0
$976$	0	0
$977$	6.91072	0.221094	0.110547	−	0.993871i	$-0.464740\pi$
$977$	6.91072	0.221094	0.110547	+	0.993871i	$0.464740\pi$
$978$	0	0
$979$	70.4141	2.25044
$980$	0	0
$981$	−7.45315	−0.237961
$982$	0	0
$983$	29.7862	0.950033	0.475017	−	0.879977i	$-0.342442\pi$
$983$	29.7862	0.950033	0.475017	+	0.879977i	$0.342442\pi$
$984$	0	0
$985$	−15.9703	−0.508855
$986$	0	0
$987$	−16.5720	−0.527491
$988$	0	0
$989$	−63.6820	−2.02497
$990$	0	0
$991$	−19.6830	−0.625252	−0.312626	−	0.949876i	$-0.601209\pi$
$991$	−19.6830	−0.625252	−0.312626	+	0.949876i	$0.601209\pi$
$992$	0	0
$993$	36.5963	1.16135
$994$	0	0
$995$	12.0075	0.380664
$996$	0	0
$997$	19.3853	0.613940	0.306970	−	0.951719i	$-0.400685\pi$
$997$	19.3853	0.613940	0.306970	+	0.951719i	$0.400685\pi$
$998$	0	0
$999$	−40.5075	−1.28160

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3380.2.a.r.1.3	✓	9
13.5	odd	4		3380.2.f.j.3041.6		18
13.8	odd	4		3380.2.f.j.3041.5		18
13.12	even	2		3380.2.a.s.1.3	yes	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3380.2.a.r.1.3	✓	9	1.1	even	1	trivial
3380.2.a.s.1.3	yes	9	13.12	even	2
3380.2.f.j.3041.5		18	13.8	odd	4
3380.2.f.j.3041.6		18	13.5	odd	4

Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3380.a (trivial)

\(f(q)\)	\(=\)	\(q-1.40883 q^{3} -1.00000 q^{5} +2.17051 q^{7} -1.01520 q^{9} +O(q^{10})\)	\(q-1.40883 q^{3} -1.00000 q^{5} +2.17051 q^{7} -1.01520 q^{9} +5.73367 q^{11} +1.40883 q^{15} -5.24468 q^{17} +7.00804 q^{19} -3.05787 q^{21} -8.19819 q^{23} +1.00000 q^{25} +5.65673 q^{27} +9.54822 q^{29} +2.41917 q^{31} -8.07777 q^{33} -2.17051 q^{35} -7.16094 q^{37} -7.16158 q^{41} +7.76780 q^{43} +1.01520 q^{45} +5.41944 q^{47} -2.28890 q^{49} +7.38886 q^{51} +3.94223 q^{53} -5.73367 q^{55} -9.87313 q^{57} +0.993988 q^{59} -4.28702 q^{61} -2.20350 q^{63} -11.6322 q^{67} +11.5499 q^{69} -5.19073 q^{71} +3.06083 q^{73} -1.40883 q^{75} +12.4450 q^{77} +7.33131 q^{79} -4.92377 q^{81} -2.86146 q^{83} +5.24468 q^{85} -13.4518 q^{87} +12.2808 q^{89} -3.40820 q^{93} -7.00804 q^{95} +7.35295 q^{97} -5.82083 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - q^{3} - 9 q^{5} - q^{7} + 12 q^{9}+O(q^{10}) \) 9 * q - q^3 - 9 * q^5 - q^7 + 12 * q^9	\( 9 q - q^{3} - 9 q^{5} - q^{7} + 12 q^{9} + 7 q^{11} + q^{15} + 13 q^{17} + 4 q^{19} + 3 q^{21} + 12 q^{23} + 9 q^{25} - 4 q^{27} + 16 q^{29} - 13 q^{31} - 34 q^{33} + q^{35} - q^{37} + 6 q^{41} + q^{43} - 12 q^{45} + 2 q^{47} + 20 q^{49} + 11 q^{51} + 30 q^{53} - 7 q^{55} - 38 q^{57} - 15 q^{59} + 21 q^{61} + 17 q^{63} + 7 q^{67} + 15 q^{69} + 7 q^{71} + 28 q^{73} - q^{75} + 46 q^{77} + 31 q^{79} + 41 q^{81} - 45 q^{83} - 13 q^{85} + 28 q^{87} + 41 q^{89} + 11 q^{93} - 4 q^{95} - 8 q^{97} + 81 q^{99}+O(q^{100}) \) 9 * q - q^3 - 9 * q^5 - q^7 + 12 * q^9 + 7 * q^11 + q^15 + 13 * q^17 + 4 * q^19 + 3 * q^21 + 12 * q^23 + 9 * q^25 - 4 * q^27 + 16 * q^29 - 13 * q^31 - 34 * q^33 + q^35 - q^37 + 6 * q^41 + q^43 - 12 * q^45 + 2 * q^47 + 20 * q^49 + 11 * q^51 + 30 * q^53 - 7 * q^55 - 38 * q^57 - 15 * q^59 + 21 * q^61 + 17 * q^63 + 7 * q^67 + 15 * q^69 + 7 * q^71 + 28 * q^73 - q^75 + 46 * q^77 + 31 * q^79 + 41 * q^81 - 45 * q^83 - 13 * q^85 + 28 * q^87 + 41 * q^89 + 11 * q^93 - 4 * q^95 - 8 * q^97 + 81 * q^99

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